Tran Le Thai^a* , D.V. Tarlakovskii^a;b**
^aMoscow Aviation Institute (National Research University), Moscow, 125993 Russia
^bResearch Institute of Mechanics, Moscow State University, Moscow, 119192 Russia
E-mail: *tranlethaivvk@gmail.com, **tdvhome@mail.ru
Received February 28, 2017

Full text PDF

Abstract

  An elastic homogeneous isotropic half-space filled with the Cosserat medium has been considered. The deformed state is characterized by independent displacement and rotation vectors. At the initial instant of time and at infinity, there are no perturbations. On the boundary of a half-space, normal displacements have been given. All components of the stress-strain state are supposed to be limited. A cylindrical coordinate system with an axis directed inward into the half-space has been used. With allowance for axial symmetry, the resolving system of equations includes three hyperbolic equations with respect to the scalar potential and the nonzero components of the vector potential and the rotation vector. The components of displacement vectors, rotation angle, stress tensors, and stress moments are related to the potentials by the known relationships.

 The solution of the problem has been sought in the form of generalized convolutions of the given displacement with the corresponding surface influence functions. To construct the latter, Hankel transforms along the radius and Laplace transforms in time have been applied. All images have been presented in three terms. The first of them correspond to the tension-compression wave, and the other two are determined by the associated shear and rotation waves. The originals of the first components have been found accurately by means of successive reversal of the transformations. For the remaining terms, we have used expansion in power series in a small parameter characterizing the connection between the shear and rotation waves. The images of the first two coefficients of these series have been found. The corresponding originals have been determined by the successive inversion of the transformations.

 Examples of calculations of the regular components of the influence of a granular composite from an aluminum shot in an epoxy matrix have been given.

Keywords: Cosserat medium, superficial influence function, Laplace and Hankel transforms, small parameter method, relation between flat and axially symmetric problems

 Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 15-08-00787).

References

1. Badriev I.B, Banderov V.V., Makarov M.V. Mathematical simulation of the problem of the pre-critical sandwich plate bending in geometrically nonlinear one dimensional formulation. IOP Conf. Ser.: Mater. Sci. Eng., 2017, vol. 208, no. 1, art. 012002, pp. 1-7. doi: iopscience.iop.org/1757-899X/208/1/012002.

2. Badriev I.B., Makarov M.V., Paimushin V.N. Contact statement of mechanical problems of reinforced on a contour sandwich plates with transversally-soft core. Russ. Math., 2017, vol. 61, no. 1, pp. 69-75. doi: 10.3103/S1066369X1701008X.
3. Badriev I.B., Makarov M.V., Paimushin V.N. Numerical investigation of a physically nonlinear problem of longitudinal bending of sandwich plate with transversal-soft core. PNRPU Mech. Bull., 2017, no. 1, pp. 39-51. (In Russian)
4. Badriev I.B., Makarov M.V., Paimushin V.N. Longitudinal and transverse bending by a cylindrical shape of the sandwich plate stiffened in the end sections by rigid bodies. IOP Conf. Ser.: Mater. Sci. Eng., 2016, vol. 158, no. 1, art. 012011, pp. 1-9. doi: iopscience.iop.org/10.1088/1757-899X/158/1/012011.
5. Badriev I.B., Makarov M.V., Paimushin V.N. Numerical investigation of physically non-linear problem of sandwich plate bending. Proc. Eng., 2016, vol. 150, pp. 1050-1055. doi: 10.1016/j.proeng.2016.07.213.
6. Badriev I.B., Makarov M.V., Paimushin V.N. Mathematical simulation of nonlinear problem of three-point composite sample bending test. Proc. Eng., 2016, vol. 150, pp. 1056-1062. doi: 10.1016/j.proeng.2016.07.214.
7. Badriev I.B., Garipova G.Z., Makarov M.V., Paymushin V.N. Numerical solution of the issue about geometrically nonlinear behavior of sandwich plate with transversal soft filler. Res. J. Appl. Sci., 2015, vol. 10, no. 8, pp. 428-435. doi: 10.3923/rjasci.2015.428.435.
8. Cosserat E., Cosserat F. Theorie des corps deformables. Paris, Librairie Scientifique A. Hermann et Fils, 1909. 226 p. (In French)
9. Aero EL, Kuvshinskiy E.V. Basic equations of the theory of elasticity of media with rotational interaction of particles. Fiz. Tverd. Tela, 1960, vol. 2, no. 7, pp. 1399-1409. (In Russian)
10. Novatsky V. Theory of Elasticity. Moscow, Mir, 1975. 872 p. (In Russian)
11. Birsan M. Several results in the dynamic theory of thermoelastic Cosserat shells with voids. Mech. Res. Commun., 2006, vol. 33, no. 2, pp. 157-176. doi: 10.1016/j.mechrescom.2005.08.008.
12. Birsan M. Thermal stresses in cylindrical Cosserat elastic shells. Eur. J. Mech., A: Solids, 2009, vol. 28, no. 1, pp. 94{101. doi: 10.1016/j.euromechsol.2008.03.001.
13. Kumar R., Gupta R.R. Propagation of waves in transversely isotropic micropolar generalized thermoelastic half space. Int. Commun. Heat Mass Transfer, 2010, vol. 37, no. 10, pp. 1452-1458. doi: 10.1016/j.icheatmasstransfer.2010.08.001.
14. Nistor I. Generalized theory of Cosserat thermoelastic media. Bull. Inst. Polytech. Jassy, 1991, vol. 37, no. 1, pp. 89-96.
15. Kulesh M.A., Matveenko V.P., Shardakov I.N. On the properties of surface waves in the elastic Cosserat medium. Mat. Model. Sist. Protsessov. Perm, PGTU, 2006, no. 14, pp. 109-113. (In Russian)
16. Suvorov E.M., Tarlakovskii D.V., Fedotenkov G.V. The plane problem of the impact of a rigid body on a half-space modelled by a Cosserat medium. J. Appl. Math. Mech., 2012, vol. 76, no. 5, pp. 511-518. doi: 10.1016/j.jappmathmech.2012.11.015.
17. Lai Thanh Tuan, Tarlakovsky D.V. Propagation of non-stationary kinematic perturbations from a spherical cavity in the Cosserat pseudocontinuum. Mekh. Kompoz. Mater. Konstr., 2011, vol. 17, no. 2, pp. 184-195. (In Russian)
18. Lai Thanh Tuan, Tarlakovsky D.V. Propagation of nonstationary axisymmetric perturbations from the surface of a sphere filled with a Cousser pseudoelastic medium. Tr. Mosk. Aviats. Inst., 2012, no. 53. Available at: http://trudymai.ru/published.php?ID=29267/. (In Russian)

19. Lai Thanh Tuan, Tarlakovsky D.V. Diffraction of waves by a spherical cavity in the Cosserat pseudocontinuum. Radioelectron., Nanosist., Inf. Tekhnol., 2013, vol. 5, no. 1, pp. 119{125. (In Russian)
20. Van Der Pol B., Bremmer H. Operational Calculus based on the Two-Sided Laplace Transform. Cambridge Univ. Press, 1950. 415 p.
21. Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Waves in Continuous Media. Moscow, FIZMATLIT, 2004. 472 p. (In Russian)
22. Dech G. Guide to the Practical Use of the Laplace Transform and z-Transform. Moscow, Nauka, 1971. 288 p. (In Russian)
23. Sagomonyan A.Ya. Stress Waves in Continuous Media. Moscow, Izd. Mosk. Gos. Univ., 1985. 416 p. (In Russian)
24. Slepyan L.I., Yakovlev Yu.S. Integral Transformations in Unsteady Problems of Mechanics. Leningrad, Sudostroyeniye, 1980. 344 p. (In Russian)
25. Sneddon I. Fourier Transforms. Courier Corp., 1995. 542 p.
26. Erofeev V.I. Wave Processes in Solids with Microstructure. Moscow, Izd. Mosk. Gos. Univ., 1999, 328 p. (In Russian)

For citation: Tran Le Thai, Tarlakovskii D.V. Nonstationary axisymmetric motion of an elastic momentum semi-space under non-stationary normal surface movements. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 2, pp. 231–245. (In Russian)

The content is available under the license Creative Commons Attribution 4.0 License.